Abstract

We discuss motions of an elastic N × M membrane model whose constituents can bind reversibly with strength ɛ to adhesive sites of a flat substrate. One of the edges of the membrane (“front”) is driven in one direction at rate constant p by N stochastically treadmilling short parallel lines (“cortex”). The main conclusions derived from Monte Carlo studies of this model are the following: (a) Since the polymerizing cortex pushes only the leading edge of the membrane, the major part of the membranes is dragged behind. Therefore, the locomotion of the membrane can be described by frictional sliding processes which are asymmetrically distributed between front and rear of the membrane. A signature of this asymmetry is the difference between the life times of adhesion bonds at front and rear, τ1 and τM, respectively, where τ1 ≫ τM. (b) There are four characteristic times for the membrane motion: The first time, T0 ∼ τM ∼ eaɛ, is the resting time where the displacement of the membrane is practically zero. The second time, Tp ∼ τ1 ∼ M, is the friction time which characterizes the time between two consecutive ruptures of adhesion bonds at the front, and which signalizes the onset of drift (“protrusion”) at the leading edge. The third time, Tr ∼ Mγ(ɛ) (γ > 1), characterizes the “retraction” of the trailing edge, which is the retarded response to the pulling leading edge. The fourth time, TL ∼ M2, is the growth time for fluctuation of the end-to-end distance. (c) The separation of time scales, Tr/Tp ∼ Mγ(ɛ) − 1, leads to stretched fluctuations of the end-to-end distance, which are considered as stochastic cycles of protrusion and retraction on the time scale of TL. (d) The drift velocity v obeys anomalous scaling, , where f (z) ∼ const. for small drag pM ≪ 1, and f (z) ∼ z−γ(ɛ) for pM ≫ 1, which implies . These results may also turn out to be useful for the (more difficult) problem of understanding the protrusion-retraction cycle of crawling biological cells. We compare our model and our results to previous two-particle theories for membrane protrusion and to known stochastic friction models.